Properties

Label 2-176-1.1-c1-0-1
Degree $2$
Conductor $176$
Sign $1$
Analytic cond. $1.40536$
Root an. cond. $1.18548$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 2·7-s − 2·9-s − 11-s + 4·13-s + 15-s − 2·17-s + 2·21-s + 23-s − 4·25-s − 5·27-s − 7·31-s − 33-s + 2·35-s + 3·37-s + 4·39-s − 8·41-s + 6·43-s − 2·45-s − 8·47-s − 3·49-s − 2·51-s − 6·53-s − 55-s − 5·59-s + 12·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.755·7-s − 2/3·9-s − 0.301·11-s + 1.10·13-s + 0.258·15-s − 0.485·17-s + 0.436·21-s + 0.208·23-s − 4/5·25-s − 0.962·27-s − 1.25·31-s − 0.174·33-s + 0.338·35-s + 0.493·37-s + 0.640·39-s − 1.24·41-s + 0.914·43-s − 0.298·45-s − 1.16·47-s − 3/7·49-s − 0.280·51-s − 0.824·53-s − 0.134·55-s − 0.650·59-s + 1.53·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $1$
Analytic conductor: \(1.40536\)
Root analytic conductor: \(1.18548\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 176,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.458816616\)
\(L(\frac12)\) \(\approx\) \(1.458816616\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.95575045127536336185363327794, −11.50606278511501469560040413868, −10.88023677523055899886301116177, −9.531323384279074079649336150802, −8.595888941067365837350463793393, −7.81893826163275369649161903506, −6.29323672924194667424631097798, −5.16796004190121525070885092557, −3.57632858228471856718550416679, −2.01168702564605014700712887669, 2.01168702564605014700712887669, 3.57632858228471856718550416679, 5.16796004190121525070885092557, 6.29323672924194667424631097798, 7.81893826163275369649161903506, 8.595888941067365837350463793393, 9.531323384279074079649336150802, 10.88023677523055899886301116177, 11.50606278511501469560040413868, 12.95575045127536336185363327794

Graph of the $Z$-function along the critical line